Summary: to appear in Math. Proc. Camb. Phil. Soc.
Spotting Infinite Groups
By Daniel Allcocky
Department of Mathematics, University of Utah, Salt Lake City, UT 84102
(Received 6 December 1996; revised 2 May 1997)
We generalize a theorem of R. Thomas, which sometimes allows one to tell by inspection that a finitely
presented group G is infinite. Groups to which his theorem applies have presentations with not too many
more relators than generators, with at least some of the relators being proper powers. Our generalization
provides lower bounds for the ranks of the abelianizations of certain normal subgroups of G in terms of their
indices. We derive Thomas's theorem as a special case.
There is a very simple theorem which states that any group with a presentation with at least one more
generator than relations is infinite. To see this, one simply abelianizes the group and uses linear algebra to
conclude that the abelianization is infinite. It is also classical that the (p; q; r) triangle group, given by the
ha; b j 1 = a p = b q = (ab) r i;
is infinite when 1=p+1=q+1=r Ÿ 1. One sees this by realizing the triangle group as a group of transformations
of the Euclidean plane or the hyperbolic plane, and simply observing that the group is infinite. (See .)